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Let $\lambda\in\mathbb{R}^n$ be a vector with positive components. Denote by $\vec{1}\in\mathbb{R}^n$ the vector with each component equal to $1$.

I am seeking to simplify the following sum for any fixed $i$:

$$ \sum_{\substack{G\in\{0,1\}^n\\G_i=1}} (-1)^{G^\top\vec{1}}\frac{\lambda_i}{G^\top\lambda}. $$

The sum is taken over all boolean vectors $G$ of size $n$ with the constraint that the $i$th component of $G$ must always be $1$. The $i$th component of $\lambda$ is denoted $\lambda_i$.

By the term "simplify", I mean find a way of computing it exactly with a computational complexity as low as possible. I am seeking a more efficient way of computing the sum than simply summing over all $2^{n-1}$ vectors $G$.

Tullio
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    Not really simpler, just an alternative way to write it in case it helps somebody: $$\lambda_i \sum_{k=1}^n (-1)^k \sum_{\substack{S \subseteq{1,\dots,n}:\ |S|=k, S \ni i}} \frac{1}{\sum_{j \in S} \lambda_j}$$ – RobPratt May 27 '20 at 17:47

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